| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 2.61·5-s − 1.61·6-s − 2.85·7-s − 2.23·8-s − 2·9-s − 4.23·10-s + 11-s − 0.618·12-s + 3.47·13-s − 4.61·14-s + 2.61·15-s − 4.85·16-s + 0.236·17-s − 3.23·18-s − 2.76·19-s − 1.61·20-s + 2.85·21-s + 1.61·22-s + 1.23·23-s + 2.23·24-s + 1.85·25-s + 5.61·26-s + 5·27-s − 1.76·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 1.17·5-s − 0.660·6-s − 1.07·7-s − 0.790·8-s − 0.666·9-s − 1.33·10-s + 0.301·11-s − 0.178·12-s + 0.962·13-s − 1.23·14-s + 0.675·15-s − 1.21·16-s + 0.0572·17-s − 0.762·18-s − 0.634·19-s − 0.361·20-s + 0.622·21-s + 0.344·22-s + 0.257·23-s + 0.456·24-s + 0.370·25-s + 1.10·26-s + 0.962·27-s − 0.333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 0.0901T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 0.0901T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 5.32T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.56426593197010668947299479173, −10.47412010902357538531671035869, −9.091535230729964662820616315035, −8.265319807331788902528017453824, −6.71722187275342871662693756837, −6.12483968027025517708425791360, −4.98762802351003999433345023780, −3.83340761680378082441701435865, −3.16319960542823909784227014131, 0,
3.16319960542823909784227014131, 3.83340761680378082441701435865, 4.98762802351003999433345023780, 6.12483968027025517708425791360, 6.71722187275342871662693756837, 8.265319807331788902528017453824, 9.091535230729964662820616315035, 10.47412010902357538531671035869, 11.56426593197010668947299479173
